1D Smoothing

This example shows how to use the pylops.Smoothing1D operator to smooth an input signal along a given axis.

Derivative (or roughening) operators are generally used regularization in inverse problems. Smoothing has the opposite effect of roughening and it can be employed as preconditioning in inverse problems.

A smoothing operator is a simple compact filter on lenght \(n_{smooth}\) and each elements is equal to \(1/n_{smooth}\).

import matplotlib.pyplot as plt
import numpy as np

import pylops

plt.close("all")

Define the input parameters: number of samples of input signal (N) and lenght of the smoothing filter regression coefficients (\(n_{smooth}\)). In this first case the input signal is one at the center and zero elsewhere.

N = 31
nsmooth = 7
x = np.zeros(N)
x[int(N / 2)] = 1

Sop = pylops.Smoothing1D(nsmooth=nsmooth, dims=[N], dtype="float32")

y = Sop * x
xadj = Sop.H * y

fig, ax = plt.subplots(1, 1, figsize=(10, 3))
ax.plot(x, "k", lw=2, label=r"$x$")
ax.plot(y, "r", lw=2, label=r"$y=Ax$")
ax.set_title("Smoothing in 1st direction", fontsize=14, fontweight="bold")
ax.legend()

Out:

<matplotlib.legend.Legend object at 0x7f90da42ca58>

Let’s repeat the same exercise with a random signal as input. After applying smoothing, we will also try to invert it.

N = 120
nsmooth = 13
x = np.random.normal(0, 1, N)
Sop = pylops.Smoothing1D(nsmooth=13, dims=(N), dtype="float32")

y = Sop * x
xest = Sop / y

fig, ax = plt.subplots(1, 1, figsize=(10, 3))
ax.plot(x, "k", lw=2, label=r"$x$")
ax.plot(y, "r", lw=2, label=r"$y=Ax$")
ax.plot(xest, "--g", lw=2, label=r"$x_{ext}$")
ax.set_title("Smoothing in 1st direction", fontsize=14, fontweight="bold")
ax.legend()

Out:

<matplotlib.legend.Legend object at 0x7f90da251630>

Finally we show that the same operator can be applied to multi-dimensional data along a chosen axis.

A = np.zeros((11, 21))
A[5, 10] = 1

Sop = pylops.Smoothing1D(nsmooth=5, dims=(11, 21), dir=0, dtype="float64")
B = np.reshape(Sop * np.ndarray.flatten(A), (11, 21))

fig, axs = plt.subplots(1, 2, figsize=(10, 3))
fig.suptitle(
    "Smoothing in 1st direction for 2d data", fontsize=14, fontweight="bold", y=0.95
)
im = axs[0].imshow(A, interpolation="nearest", vmin=0, vmax=1)
axs[0].axis("tight")
axs[0].set_title("Model")
plt.colorbar(im, ax=axs[0])
im = axs[1].imshow(B, interpolation="nearest", vmin=0, vmax=1)
axs[1].axis("tight")
axs[1].set_title("Data")
plt.colorbar(im, ax=axs[1])
plt.tight_layout()
plt.subplots_adjust(top=0.8)